Abstract：There are two sorts of singularities: coming from colimits, we may resolve it by going higher; coming from limit, we may resolve it by going derived. In this mini-course, we introduce the derived higher Lie groupoids to host such structures. These are alternative methods than derived higher stacks and might be more straightforward/friendly for differential geometers.

Then what should be symplectic structures on such derived higher spaces? It turns out shifted symplectic structures might be the correct notion. We then further introduce deRham complex on such spaces and shifted symplectic and lagrangian structures on them. After these structures are introduced, all of sudden, classical field theory gains much space and can host higher dimensional world sheets. We will  furthermore discuss classical field theory based on shifted symplectic and lagrangian structures.  If time allows, we will come back to examples, coming from various symplectic reductions. 

Mini-course-1. 14:00-16:00 November 13, 2026

Higher Lie groupoids via Kan simplicial manifolds: They form an incomplete category of fibrant objects (iCFO). This further gives rise to an oo-category. The infinitesimal counterparts of higher Lie groupoids are higher Lie algebroids, or equivalently NQ manifolds (some sort of differential graded manifold with non-negative gradings). 

 

Mini-course-2. 14:00-16:00 November 20, 2026

We introduce derived manifolds: They form a CFO; and  derived higher Lie groupoids:  They form an iCFO. 

Mini-course-3. 14:00-16:00 November 27, 2026

deRham forms on derived higher Lie groupoids. Then we introduce shifted symplectic forms and shifted lagrangian structures. 

Mini-course-4. 14:00-16:00 December 4, 2026

classical field theory picture and various symplectic reduction in the world of shifted symplectic derived higher groupoids.